For what value(s) of k does the system have....?

i) For what value(s) of k does the system have;

no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3

0x + y - k-3 = -5

0x + 0y + k^2 -2k = 5k + 11

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k. (Crying)

Re: For what value(s) of k does the system have....?

Quote:

Originally Posted by

**figleaf7** i) For what value(s) of k does the system have;

no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3

0x + y - k-3 = -5

0x + 0y + k^2 -2k = 5k + 11

No wonder you are confused- either you copied this incorrectly or it was given badly. I suspect it was supposed to be

$\displaystyle x+ 2y- z= -3$

$\displaystyle y- (k-3)z= -5$

$\displaystyle (k^2- 2k)z= 5k+ 11$

If $\displaystyle k^2- 2k\ne 0$ then $\displaystyle z= \frac{5k+ 11}{k^2- 2k}$

Now what happens if $\displaystyle k^2- 2k= 0$ but $\displaystyle 5k+ 11\ne 0$? What if both are 0?

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k. (Crying)[/QUOTE]

Re: For what value(s) of k does the system have....?

Quote:

Originally Posted by

**figleaf7** i) For what value(s) of k does the system have;

no solutions, a unique solution, infinitely many solutions?

x + 2y - z = -3

0x + y - k-3 = -5

0x + 0y + k^2 -2k = 5k + 11

ii) each of these equations represents a plane. In each case in i) give a geometric description of the intersection of the three planes.

I have answered this already, but have gotten myself hopelessly confused, and the more I spend time on it, the worse I get. PLEASE HELP. I just can't seem to work out how to find the values of k. (Crying)

In matrix form this system is

$\displaystyle \begin{align*} \left[ \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix}\right] \left[ \begin{matrix} x \\ y\\ z \end{matrix} \right] &= \left[ \begin{matrix} -3 \\ -5 \\ \phantom{-} 11 \end{matrix} \right] \end{align*}$

This matrix equation has a unique solution where $\displaystyle \begin{align*} \left| \begin{matrix} 1 & 2 & -1 \\ 0 & 1 & -k-3 \\ 0 & 0 & k^2 - 7k \end{matrix} \right| \neq 0 \end{align*}$.

It will either have no solution or infinite solutions where the determinant IS 0, so once you know these k values you will then need to investigate the system with those values plugged in.

Re: For what value(s) of k does the system have....?

Yes, you're right, I copied it poorly. Sorry, was on night shift.